IMPACT PARAMETER INTERPRETATION OF PROTON- PROTON ...

Nuclear Physics B166 (1980) 301-320 © North-Holland Publishing Company

I M P A C T P A R A M E T E R I N T E R P R E T A T I O N OF P R O T O N PROTON SCATTERING FROM A CRITICAL REVIEW OF ALL ISR D A T A U. A M A L D I CERN, Geneva, Switzerland

K.R. S C H U B E R T Institut [fir Hochenergiephysik, Universitdt Heidelberg, Germany Received 10 December 1979

This paper contains a critical review of all the data produced at the ISR on proton-proton elastic scattering and total cross sections. This coherent and complete set of data is used to compute the impact parameter distribution of the proton-proton inelastic overlap integral. This impact parameter analysis has smaller errors than any other previously made, and confirms the good agreement with the geometrical scaling model while strongly disagreeing with models based on factorizing eikonals. For the first time we find indications of a second contribution to the peripheral rising of the proton-proton cross section in a region around 2.2 fm.

1. Introduction In 1971, a few months after the successful start-up of the C E R N Intersecting Storage Rings (ISR) [1], the first data on elastic scattering were obtained. In the following eight years, various groups measured differential cross sections, total cross sections, and real parts of the forward nuclear amplitude at the ISR. All these groups have published their results and no new information is expected to come from the ISR in the next years. Having been personally involved in m a n y of these experimens, we felt the need to conclude this extensive p r o g r a m of m e a s u r e m e n t s by producing a coherent picture of the p h e n o m e n a under study. For this reason the present p a p e r contains a critical review of the data, which takes into account the quality of the various experiments by eliminating the (unavoidable) contradictions through a careful consideration of the sources of systematic errors. The numerical results of such an analysis, with all the relevant tables of cross sections, are to be published by one of us (K.R.S.) in the numerical tables of the series of Landolt-B6rnstein [2]. In sect. 2 of the present p a p e r we discuss the available experiments and the choices made in their compilation. The output of this work represents the most coherent set of data on p r o t o n - p r o t o n scattering available in the largest and (up till now) highest energy range. This allows us to complete a program that was first started in 1973 by one of us (U.A.), 301

U. Amaldi, K. R. Schubert / Review of all ISR data

302

immediately after the discovery of the rising proton-proton cross sections [3,4]. By computing the impact parameter distribution of the proton-proton overlap integral with the few data then available, it was possible to conclude that the rise of the total cross section in the ISR energy range is a peripheral phenomenon [5]. This result contradicted the only previous analysis [6]. The first analysis was soon followed by others [7-lo] that confirmed the result and led to the application of the very fruitful idea of geometrical scaling to proton-proton scattering [ll]. With the complete and coherent set of data now at our disposal, we have performed a new overlap-integral analysis, which is presented in sect. 3 together with the discussion of the propagation of the statistical and systematic errors. Sect. 4 summarizes the results and compares them with the ones obtained previously. 2. Compilation

of the data

2.1. Total cross sections

After the first experiments, which discovered that total proton-proton cross sections rise by about 10% in the ISR energy range [3,4], three other measurements were published [12-141. All the published data agree within the errors (see fig. 13 of ref. [14]). The data of the CERN-Pisa-Rome-Stony Brook Collaboration [13] were obtained by the authors of refs. [3,4] and, being much more accurate, supersede their previous results. For this reason table 1 reproduces only the final data of refs. [12-141. Note that in experiments [13] and [14] three different methods were applied to obtain crtot: (i) extrapolation of the elastic rate to zero angle and measurement of the luminosity by means of the Van der Meer method [15] (this is the so-called CERN-Rome method); (ii) measurement of the total interaction rate and of the luminosity (Pisa-Stony Brook method); (iii) luminosity-independent method, which combines the elastic rate with the total interaction rate. In ref. [12] only the second method was used, and this explains the larger errors. In the experiments of refs. [13, 143 the largest contribution to the scale error (i.e., to the error which is equal at all TABLE 1 Total proton-proton

cross sections

\

~t,,W)

Js (GeV) \ 23.5 30.7 44.7 52.8 62.5 Scale error

From ref. [12]

From ref. [13]

From ref. [14]

Average

38.10 f 0.70 40.00 f 0.60 42.5OztO.80 42.9OkO.70 44.10*0.90

38.89 f 0.21 40.17*0.21 41.66zt0.19 42.46 * 0.26 43.04kO.29

39.1*0.3 40.1* 0.3 42.0 * 0.3 42.9k0.3 43.7*0.4

38.94kO.17 40.14*0.17 41.79ztO.16 42.67kO.19 43.32 k0.23

+z0.28

20.32

\

f 0.25

U. Amaldi, K. R. Schubert/Review of all 1SR data

303

TABLE 2

Ratio 0 between the real and the imaginary part of the forward nuclear amplitude ~s(GeV)

p

Ref.

23,5 30.7 44.7 52.8 62.5

0.02 ±0.05 0.042+0.011 0.062+0.011 0.078+0.010 0.095+0.011

[16] [17] [17] [17] [17]

Scale error

+ 0.015

energies) comes from the error in the step size of the vertical displacement of the beams used in the Van der M e e r method. The errors in the experiments are thus strictly correlated, and the averaging procedure hardly implies a reduction of the scale error.

2.2. R e a l part of the forward nuclear amplitude Two experiments have been p e r f o r m e d by the C E R N - R o m e Collaboration to measure the ratio p between the real and the imaginary part of the forward nuclear amplitude [16, 17] by detecting the interference effect between Coulomb and nuclear scattering. The experiments found a destructive interference that increases with energy and, through the use of a dispersion relation, have allowed the authors to reach the conclusion that the p r o t o n - p r o t o n total cross section rises continuously at least up to a laboratory m o m e n t u m of about 4 0 T e V / c [17]. The results are summarized in table 2.

2.3. Differential cross sections In the compilation of the differential cross sections we have used the data of all the experiments [14, 17, 18-22] listed in table 3. At each of the five standard ISR energies, we have normalized all the available data to the optical point and relative to each other by taking into account both the statistics and the systematic errors. The results of this procedure are summarized in table 4. We distinguish between "normalized", "unnormalized", "interpolated" and " e x t r a p o l a t e d " data. " N o r m a l i z e d " data are given in the original p a p e r in m b / ( G e V / c ) 2 and are accompanied by a scale error. In the course of our procedure we readjust their normalization according to the general behaviour of the differential cross section, and scale them, if necessary, by interpreting the scale error as a 1 standard deviation value. (The scaling factors are given in column 5 of table 4.)

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U. ArnaMi, K. R. Schubert/Review of all ISR data

TABLE 3 Sources of differential proton-proton cross sections Papers (as indicated in ref. [2])

Ref.

Energies ~/s (GeV)

72B2

[18]

23.5, 30.7,44.7, 52.8

74B1 75K1

[19] [20]

23.5,30.7,44.7,52.8 23.5, 62.5

76A2 77A2

[21] [17]

23.5, 30.7, 32.4, 35.2, 38.3 30.7,44.7, 52.8,62.5

78B1

[14]

44.7, 52.8,62.5

78N1

[22]

23.5, 30.7,44.7, 52.8,62.5

Remarks The data are unnormalized and are based on a subset of the published data (private information from P. Strolin) The data are not normalized The t-range is 0.21-0.8 (GeV/c) 2. Above 0.8 (GeV/c) 2 the data are replaced by those of 78N1 Normalized to ±10% These data are in the Coulomb interference region. At x/s = 62.5 they extend further The data correspond to the published figure (private communication from A. BShm) The data are in the large t-region: Itl > 0.8 (GeV/c) 2

T h e " u n n o r m a l i z e d " data are given, in the original paper, in n u m b e r of events per bin; by joining t h e m with the data at smaller and larger m o m e n t u m transfers we obtain the normalization constants of c o l u m n 6 in table 4. In this case the scale error of each set of data (column 7 of table 4) is estimated on the basis of the scale errors of the contiguous sets. W h e n two experiments overlap, we take the data f r o m the e x p e r i m e n t that has the smaller errors. In s o m e t-ranges there are no data and we interpolate t h e m b e t w e e n the data of the two contiguous sets. For the present analysis we forget a b o u t the C o u l o m b interference region and include only data points a b o v e Itl = 0.01 ( G e V / c ) 2. T h e data in this gap are o b t a i n e d by interpolation b e t w e e n the optical point and the first set of data using the slope b of the f o r w a r d nuclear cross section, usually written for small m o m e n t u m transfers in the f o r m do- = (d_~) e_blt,. dt 0

(1)

By following the p r o c e d u r e of ref. [13], we use for b a linearly interpolated value in In s b e t w e e n the slope data m e a s u r e d at the largest F N A L energies [23] and the I S R m e a s u r e m e n t s at large energies [18, 24]. T h e scale error of these " i n t e r p o l a t e d " data is o b t a i n e d by assuming an uncertainty zlb = ±0.3 ( G e V / c ) -2. In the c o m p u t a t i o n s to be described in sect. 3, this is the only error attributed to these fake data, since the points are taken to lie on the interpolated curves w i t h o u t statistical errors. A t m o m e n t u m transfers larger than 3.0 ( G e V / ~ ) 2 we " e x t r a p o l a t e " the cross section 2 with an e n e r g y - i n d e p e n d e n t slope [22] of 1.8 5 ( G e V / c ) - . This is l e g i t i m a t e b e c a u s e this t-region gives a very small contribution to all the results of our analysis.

305

U. Amaldi, K. R. Schubert/Review of all ISR data TABLE 4 Compilation of the data on the proton-proton differential cross section

(GeV)

23.5

30,7

44.7

52.8

62.5

t-range (GeV/c) 0.0-0.04 0.04-0.24 0.24-0.80 0.80-3.0 3.0-6.0 0.0-0.015 0.015-0.112 0.112-0.458 0.458-0.88 0.88-1.16 1.16-3.0 3.0-6.0 0.0-0.009 0,009-0.02 0,02-0.053 0.053-0.29 0.29-0.60 0.60-1.16 1.16-3.0 3.0-6.0 0.0-0.009 0.009-0.018 0.018-0.03 0.03-0.075 0.075-0.45 0.45-0.72 0.72-1.27 1.27-3.0 3.0-6.0 0.0-0.009 0.009-0.036 0.036-0.10 0.10-0.265 0.265-0.8 0.8-3.0 3.0-6.0

Paper

Type of data

72B2 74B1 78N1

interpolated unnormalized unnormalized normalized extrapolated interpolated unnormalized unnormalized unnormalized averaged normalized extrapolated

72B2 72B2 74B1 78N1

interpolated normalized normalized unnormalized interpolated unnormalized normalized extrapolated

77A2 78B1 72B2 74B1 78N1

interpolated normalized interpolated normalized unnormalized interpolated unnormalized normalized extrapolated

77A2 78B1 72B2 74B1 78N1

interpolated normalized normalized interpolated normalized normalized extrapolated

77A2 78B1 75K1 78N1

Scaling factors

Normaliz. const, (nb/event)

32700 102 1.075

12700 7760 20.4 1.000

1.006 0.990 8200 1,3 1.000

1.010 0.990 4620 0.72 1.030

1.010 0.980 1.000 1.000

Scale error (%) 1.0 1.2 3.0 5.0 10.0 0.4 0.5 2.0 3.5 5.0 5.0 10.0 0.25 1.0 1.0 2.0 3.5 5.0 5.0 10.0 0.25 1.0 1.0 1.0 2.0 3.5 5.0 5.0 10.0 0.25 1.0 1.0 3.0 5.0 5.0 10,0

W e n o w d i s c u s s e a c h e n e r g y i n t u r n , first g i v i n g t h e v a l u e o f t h e o p t i c a l p o i n t obtained from the values of equation

O'to t

a n d p a p p e a r i n g i n t a b l e s 1 a n d 2, a c c o r d i n g t o t h e

I+P 2 2 (-~t°') o - 16,n.(hc)2 o'tot.

(2)

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U. Arnaldi, K. R. Schubert/Review of all ISR data

At 2 3 . 5 G e V the forward cross section is 77.5±0.7mb/(GeV/c) 2. The unnormalized data of 72B2 start at Itl -- 0.04 ( G e V / c ) 2, and to normalize them we use the slope b of the forward nuclear cross section [13] b = 11.8±0.3 ( G e V / c ) -2. The normalization of the data of paper 72B2 turns out to be 32.7 ~b/event, and to join them smoothly with the unnormalized events of paper 74B1 these last data have to be normalized with the constant 102 nb/event. The data of paper 78N1 start at It I = 0.82 ( G e V / c ) 2, where the data of 74B1 have larger statistical errors. A smooth join is obtained by multiplying the data of paper 78N1 by 1.075, corresponding to 1.5 standard deviations of the quoted normalization uncertainty. The data of refs. [20, 21] are in agreement with the behaviour obtained by applying the described procedure. At 30.7 G e V the optical point computed from eq. (1) is 82.5 ± 0.7 mb/(GeV/c) 2. The minimum momentum transfer of the data of paper 72B2 is small [0.016 (GeV/c)2], and the normalization of the first set of data is easily found: 12.7 ixb/event. The second set of data coming from the same source [0.011 =0.8 ( G e V / c ) 2] can be used without changing the normalization. The differential cross sections at the five energies are plotted in figs. 1 and 2. As mentioned in sect. 1, the interested reader can find the numerical tables in ref. I-2]. ioo

T

,

-~,.

r

i

\

i

"X

25.5 GeV

[

"\

~

i

r

,,

50.7 GeV

X\

44.7 GeV ~"

2

A

io

X

~% ',,

A >

v

ioo

I

~X X %

I

I

l

I

L

I

I

--

62,5 GeV

52,8 GeV

k

b nD

'%

io

2... 'r

Io--0,,

[

012

015 ,

O',l I tl. (GeV/c) 2 %8

01,2

] 0.5

' O,a,

Fig. 1. Selected set of differential cross sections b e t w e e n 0 and 0.4 ( O e V / c ) 2. The selection and n o r m a l i z a t i o n p r o c e d u r e is described in the text. The errors s h o w n are statistical only. The solid lines indicate the interpolations c h o s e n in the regions w h e r e no data exist.

308

U. Amaldi, K. R. Schubert/Review o[ all ISR data !0 2 _____

I

q

v-

- f - -

23,5 GeV

\

~ - - -

-

30,7 GeV

44.7 GeV

\ L

1(3-:

L

~D

IO-~ (D v

F

:

t¢

"%÷+~,

t

~*. +*¢%

i02

tt

-S

52,8 GeV

I@-: ~-

62.5 GeV

',

÷ ÷ +

P ¢,~*•#o,~ ¢~t **%.

t

1©-6L ©

L I

2

3

o Itl

I

2

3

4

, (GeV/c) 2

Fig. 2. Selected set of differential cross sections between 0 and 4 (GeV/c) 2. The solid lines indicate interpolated values where no data exist.

2.4. Integrated elastic cross sections By numerical integration of the differential cross section we find the elastic cross sections appearing in the last column of table 5. The other columns contain the values of the elastic cross section as measured in previous experiments. The data of ref. [ 14] are systematically higher than the results of the present analysis by 0.16 mb, i.e., by o n e and a half times the scale error• (Note that the main source of scale error in both experiments c o m e s from the luminosity measurement, which means that the scale errors are strongly correlated.) By subtracting o-e~ from O'tot we get the values of the inelastic cross section Orinel reported in table 6. The quoted errors take into account the correlation between the errors on o'.l and O'tot due to the fact that our determination of o'~i contains the optical point, computed from the knowledge of O'tot.

309

U. Amaldi, K. R. Schubert / Review of all ISR data TABLET Elastic cross sections (T=,(mb) Js (GeV) 23.5 30.7 44.7 52.8 62.6

From ref. [3] (1973)

From ref. [14] (1978)

Present (1979)

6.8kO.2 7.050.2 7.5*0.3 7.6kO.3

6.82iO.08 7.39kO.08 7.45 zk0.08 7.56~tO.08 7.77zko.10

6.73 f 0.08 7.16ztO.09 7.17zto.09 7.45 * 0.09 7.66+0.11

Scale error

*o.l”’

zto.3

a) This error is estimated

analysis

* 0.09

on the basis of the error on the luminosity

quoted

in ref.

[141. TABLET Inelastic cross sections

Js (GeV

From ref. [14] (1978)

Present (1979)

23.5 30.7 44.7 52.8 62.5

32.40i0.14 32.79 * 0.22 34.55 * 0.22 35.07 * 0.20 35.79 f 0.24

32.21 ~kO.14 32.98k0.14 34.62*0.14 35.22~~0.16 35.66ztO.21

nine, (mb)

Scale error

ztO.20”’

“‘This error is estimated quoted in table 1.

analysis

zto.20

oh the basis of the error

on vttot

The ratios of gel to ctot are given in table 7. In this case, too, the strong correlations between the errors are taken into account. The ratio (+Jutot is plotted in fig. 3 versus In s together with two linear fits. The probability of the fit that has free slope is small but still acceptable (x2 = 10.0 for

TABLE? Ratios of elastic to total cross sections

9,J”,,t 4s (GeV)

From ref. [14] (1978)

Present (1979)

analysis

23.5 30.7 44.7 52.8 62.5

0.174 * 0.003 0.184*0.003 0.177 * 0.003 0.177*0.003 0.178*0.003

0.1728*0.0016 0.1784*0.0017 0.1716~0.0018 0.1746*0.0016 0.1768*0.0021

U. Amaldi, K. R. Schubert/Review of all ISR data

310

T

~

~

0180

t

~6

6--

0.172

b®

T

J

f 017C

I0

2O

40

8O

, (GeV)

Fig. 3. Ratio of the integrated elastic cross section to the total cross section as a function of energy. The errors are one standard deviation statistical errors. The dotted line indicates a ~,2_fitted straight line for O'el/O'to t 1.)eTSUS In s: its slope is compatible with zero. Imposing zero slope leads to the solid line with (trel/~rto,) = 0.1747 ± 0.0012.

N D F = 3) and the slope is compatible with zero: d O'el 0.0007±0.0040, d In s O'tot where s is measured in G e V 2. The quoted error is multiplied by x / x Z / N D F . assuming zero slope we find the ratio o'eJ O'tot= 0.1747 ± 0.0012,

By

(3)

where the error is scaled as above.

3. Overlap integrals in impact parameter space By neglecting nucleon spins, the differential cross section can be written in the form d t r / d t = If(t)l 2 = IR(t) + iA(t)l 2 ,

(4)

where R (t) and A ( t ) are the real and the imaginary parts of the scattering amplitude. At high energies the scattering angles are so small that the momentum transfer It I is the square of a vector q that lies in a plane perpendicular to the direction of the incoming protons: Itl = q2. When the incoming m o m e n t u m is larger than a few G e V / c , the scattering is mainly diffractive, i.e., it is due to the absorption of the incoming proton waves due to the many open inelastic channels. In the ISR energy range R (q)